where \(\varepsilon \)  is a primitive \(r\) th root of unity. The first resolution of the quotient singularity \(\mathbb {A}^3/G\)  to be studied was the \(G\) -Hilbert scheme \(G\text{\textnormal {-Hilb}}\,\mathbb {A}^3\) , the fine moduli space of \(G\) -clusters: zero-dimensional \(G\) -invariant subschemes \(Z\subset \mathbb {A}^3\)  with \(H^0(\mathcal {O}_Z)\cong G]\)  as \(G\) -modules. \(G\) -Hilb was shown to be smooth for abelian \(G\)  by Nakamura [1]}, and subsequently shown to be smooth – and hence a resolution – for all finite \(G\subset \operatorname{SL}_3(\)  by Bridgeland–King–Reid [2]}.
